A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary

Juan Casado-Díaz; Manuel Luna-Laynez; Francisco Javier Suárez-Grau

doi:10.1016/j.crma.2010.07.023

Partial Differential Equations/Mathematical Physics

A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
[Fluide visqueux dans un domaine de faible épaisseur vérifiant la condition de glissement sur une frontière lègèrement rugueuse]

Présenté par : Evariste Sanchez-Palencia

Juan Casado-Díaz ¹ ; Manuel Luna-Laynez ¹ ; Francisco Javier Suárez-Grau ¹

¹ Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 967-971.

Abstract (VO)
Résumé

We consider a viscous fluid of small height ε on a periodic rough bottom $Γ_{ε}$ of period $r_{ε}$ and amplitude $δ_{ε}$ , $δ_{ε} ≪ r_{ε} ≪ ε$ , where we impose the slip boundary condition. When ε tends to zero we obtain a Reynolds system depending on the limit λ of $(δ_{ε} \sqrt{ε}) / (r_{ε} \sqrt{r_{ε}})$ . If $λ = + \infty$ , the fluid behaves as if we would impose the adherence condition on $Γ_{ε}$ . This justifies why this is the usual boundary condition for viscous fluids. If $λ = 0$ the fluid behaves as if $Γ_{ε}$ was plane. Finally, for $λ \in (0, + \infty)$ it behaves as if $Γ_{ε}$ was flat but with a higher friction coefficient.

On considère un fluide visqueux de faible épaisseur ε sur un fond rugueux $Γ_{ε}$ , périodique de période $r_{ε}$ et amplitude $δ_{ε}$ , $δ_{ε} ≪ r_{ε} ≪ ε$ , où on impose la condition de glissement. Quand ε converge vers zéro on obtient un système de type Reynolds qui dépend de la limite λ de $(δ_{ε} \sqrt{ε}) / (r_{ε} \sqrt{r_{ε}})$ . Si $λ = + \infty$ , le fluide se comporte comme si on aurait imposé la condition d'adhérence sur $Γ_{ε}$ . Ceci justifie la condition usuelle pour un fluide visqueux. Si $λ = 0$ le fluide se comporte comme si $Γ_{ε}$ était plate. Enfin, pour $λ \in (0, + \infty)$ , tout se passe comme si $Γ_{ε}$ était plate, mais avec un coefficient de frottement plus élevé.

Reçu le : 2010-06-29
Accepté le : 2010-07-26
Publié le : 2010-08-11

DOI : 10.1016/j.crma.2010.07.023

Affiliations des auteurs :

Juan Casado-Díaz ¹ ; Manuel Luna-Laynez ¹ ; Francisco Javier Suárez-Grau ¹

¹ Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

@article{CRMATH_2010__348_17-18_967_0,
     author = {Juan Casado-D{\'\i}az and Manuel Luna-Laynez and Francisco Javier Su\'arez-Grau},
     title = {A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {967--971},
     publisher = {Elsevier},
     volume = {348},
     number = {17-18},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.023},
     language = {en},
}

TY  - JOUR
AU  - Juan Casado-Díaz
AU  - Manuel Luna-Laynez
AU  - Francisco Javier Suárez-Grau
TI  - A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 967
EP  - 971
VL  - 348
IS  - 17-18
PB  - Elsevier
DO  - 10.1016/j.crma.2010.07.023
LA  - en
ID  - CRMATH_2010__348_17-18_967_0
ER  -

%0 Journal Article
%A Juan Casado-Díaz
%A Manuel Luna-Laynez
%A Francisco Javier Suárez-Grau
%T A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
%J Comptes Rendus. Mathématique
%D 2010
%P 967-971
%V 348
%N 17-18
%I Elsevier
%R 10.1016/j.crma.2010.07.023
%G en
%F CRMATH_2010__348_17-18_967_0

Juan Casado-Díaz; Manuel Luna-Laynez; Francisco Javier Suárez-Grau. A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 967-971. doi : 10.1016/j.crma.2010.07.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.023/

Bibliographie
Cité par

[1] Y. Amirat; D. Bresch; J. Lemoine; J. Simon Effect of rugosity on a flow governed by stationary Navier–Stokes equations, Quart. Appl. Math., Volume 59 (2001), pp. 769-785

[2] Y. Amirat; B. Climent; E. Fernández-Cara; J. Simon The Stokes equations with Fourier boundary conditions on a wall with asperities, Math. Models Meth. Appl. Sci., Volume 24 (2001), pp. 255-276

[3] T. Arbogast; J. Douglas; U. Hornung Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., Volume 21 (1990), pp. 823-836

[4] G. Bayada; M. Chambat Homogenization of the Stokes system in a thin film flow with rapidly varying thickness, RAIRO Model. Math. Anal. Numer., Volume 23 (1989), pp. 205-234

[5] N. Benhaboucha; M. Chambat; I. Ciuperca Asymptotic behaviour of pressure and stresses in a thin film flow with a rough boundary, Quart. Appl. Math., Volume 63 (2005), pp. 369-400

[6] D. Bresch; C. Choquet; L. Chupin; T. Colin; M. Glisclon Roughness-induced effect at main order on the Reynolds approximation, Multiscale Model. Simul., Volume 8 (2010), pp. 997-1017

[7] D. Bucur; E. Feireisl; N. Nečsová Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions, Arch. Rational Mech. Anal., Volume 197 (2010), pp. 117-138

[8] D. Bucur; E. Feireisl; S. Nečasová; J. Wolf On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries, J. Differential Equations, Volume 244 (2008), pp. 2890-2908

[9] J. Casado-Díaz Two-scale convergence for nonlinear Dirichlet problems in perforated domains, Proc. Roy. Soc. Edinburgh A, Volume 130 (2000), pp. 249-276

[10] J. Casado-Díaz; E. Fernández-Cara; J. Simon Why viscous fluids adhere to rugose walls: A mathematical explanation, J. Differential Equations, Volume 189 (2003), pp. 526-537

[11] J. Casado-Díaz; M. Luna-Laynez; F.J. Suárez-Grau Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall, Math. Models Meth. Appl. Sci., Volume 20 (2010), pp. 121-156

[12] D. Cioranescu; A. Damlamian; G. Griso Periodic unfolding and homogenization, C. R. Acad. Sci. Paris Ser. I, Volume 335 (2002), pp. 99-104

[13] W. Jag̈er; A. Mikelić Couette flows over a rough boundary and drag reduction, Comm. Math. Phys., Volume 232 (2003), pp. 429-455

José M. Arrieta; Jean Carlos Nakasato; Manuel Villanueva-Pesqueira Homogenization in 3D thin domains with oscillating boundaries of different orders, Nonlinear Analysis, Volume 251 (2025), p. 113667 | DOI:10.1016/j.na.2024.113667
Igor Pažanin; Francisco J. Suárez‐Grau Roughness‐induced effects on the thermomicropolar fluid flow through a thin domain, Studies in Applied Mathematics, Volume 151 (2023) no. 2, p. 716 | DOI:10.1111/sapm.12611
Matthieu Bonnivard; Igor Pažanin; Francisco J. Suárez-Grau A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 56 (2022) no. 4, p. 1255 | DOI:10.1051/m2an/2022039
Hind Al Baba; Chérif Amrouche; Miguel Escobedo Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p -Spaces, Archive for Rational Mechanics and Analysis, Volume 223 (2017) no. 2, p. 881 | DOI:10.1007/s00205-016-1048-1
Khaled El-Ghaouti Boutarene APPROXIMATE TRANSMISSION CONDITIONS FOR A POISSON PROBLEM AT MID-DIFFUSION, Mathematical Modelling and Analysis, Volume 20 (2015) no. 1, p. 53 | DOI:10.3846/13926292.2015.1000988
F.J. Suárez‐Grau Effective boundary condition for a quasi‐Newtonian viscous fluid at a slightly rough boundary starting from a Navier condition, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 95 (2015) no. 5, p. 527 | DOI:10.1002/zamm.201300160
J. Casado-Díaz; M. Luna-Laynez; F. J. Suárez-Grau Effect of Roughness on the Flow of a Fluid Against Periodic and Nonperiodic Rough Boundaries, Differential and Difference Equations with Applications, Volume 47 (2013), p. 607 | DOI:10.1007/978-1-4614-7333-6_56
J. Casado-Díaz; M. Luna-Laynez; F. J. Suárez-Grau Asymptotic Behavior of the Navier–Stokes System in a Thin Domain with Navier Condition on a Slightly Rough Boundary, SIAM Journal on Mathematical Analysis, Volume 45 (2013) no. 3, p. 1641 | DOI:10.1137/120873479
J. Casado-Díaz; M. Luna-Laynez; F. J. Suárez-Grau On the Navier boundary condition for viscous fluids in rough domains, SeMA Journal, Volume 58 (2012) no. 1, p. 5 | DOI:10.1007/bf03322603
J. Casado-Díaz; M. Luna-Laynez; F. J. Suárez-Grau Asymptotic Behavior of a Viscous Fluid Near a Rough Boundary, BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Methods, Volume 81 (2011), p. 57 | DOI:10.1007/978-3-642-19665-2_7

Cité par 10 documents. Sources : Crossref

Commentaires - Politique